Limitations of Gödel Machines

The fundamental limitations are closely related to
those first identified by Gödel's celebrated paper on
self-referential formulae [11].
Any formal system that encompasses arithmetics (or ZFC etc)
is either flawed or allows for unprovable but true statements.
Hence even a Gödel machine with unlimited computational
resources must ignore those self-improvements
whose effectiveness it cannot prove,
e.g., for lack of sufficiently powerful axioms in .
In particular, one can construct pathological
examples of environments and
utility functions that make it impossible for the machine
to ever prove a target theorem.
Compare Blum's speed-up theorem
[3,4]
based on certain incomputable predicates.
Similarly, a realistic Gödel machine with limited resources
cannot profit from self-improvements
whose usefulness it cannot prove within
its time and space constraints.

Nevertheless, unlike previous methods, it can
in principle exploit at least the provably good speed-ups
of any part of its initial software, including those
parts responsible for huge (but problem class-independent) slowdowns
ignored by the earlier approaches [16,17]
(Section 6.4).